Harmonic mean

The harmonic mean of n values is n / (1/x₁ + 1/x₂ + ... + 1/xₙ) — the reciprocal of the arithmetic mean of the reciprocals. It’s the right average when averaging rates expressed as ratio-units (miles per hour, requests per second, dollars per item).

Canonical example. Drive 60 miles at 30 mph, then 60 miles at 60 mph. What’s the average speed?

• Arithmetic mean: (30 + 60) / 2 = 45 mph. Wrong.
• Harmonic mean: 2 / (1/30 + 1/60) = 40 mph. Correct.

Sanity check: total 120 miles, total time = 60/30 + 60/60 = 2 + 1 = 3 hours. 120 / 3 = 40 mph. The harmonic mean nails it; the arithmetic mean systematically over-weights the higher rate.

The harmonic mean is always ≤ geometric mean ≤ arithmetic mean. The chain HM ≤ GM ≤ AM is one of the classical inequalities in mathematics. Choosing the wrong one is the silent cause of many real-world arithmetic errors — averaged fuel-economy figures, computed network throughput, mean-time-to-failure for parallel systems.

Use harmonic mean whenever the natural “average” is a rate over a constant numerator (constant distance, constant work to be done). Use arithmetic mean for additive quantities (heights, scores). Use geometric mean for multiplicative quantities (compound returns).